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\begin{equation*}x_1=r\cos(\phi_1)\,x_2=r\sin(\phi_1)\cos(\phi_2)\,
x_3=r\sin(\phi_1)\sin(\phi_2)\cos(\phi_3)\,\cdots\,x_{n-1}=r\sin(\phi_1)\cdots\sin(
\phi_{n-2})\cos(\phi_{n-1})\,
x_n~~\,=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\sin(\phi_{n-1})\,d^nr = \left|\det\frac{\partial (x_i)}{\partial(r,\phi_i)}\right|dr\,d\phi_1 \, d\phi_2\
ldots d\phi_{n-1}
=r^{n-1}\sin^{n-2}(\phi_1)\sin^{n-3}(\phi_2)\cdots 
\sin(\phi_{n-2})\,dr\,d\phi_1 \, d\phi_2\cdots d\phi_{n-1}V_n=\int_{r=0}^R/frac \int_{\phi_1=0}^\pi\cdots \int_{\phi_{n-2}=0}^\pi\int_{\phi_{n-1}=0}^{2\pi}d^nr. \,\end{equation*}
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